The practical implementation of Bayesian inference requires numerical approximation when closed-form expressions are not available. What types of accuracy (convergence) of the numerical approximations guarantee robustness and what types do not? In particular, is the recursive application of Bayes' rule robust when subsequent data or posteriors are approximated? When the prior is the push forward of a distribution by the map induced by the solution of a PDE, in which norm should that solution be approximated? Motivated by such questions, we investigate the sensitivity of the distribution of posterior distributions (i.e. posterior distribution-valued random variables, randomized through the data) with respect to perturbations of the prior and data generating distributions in the limit when the number of data points grows towards infinity.